In mathematics, an axiom of countability is a property of certain mathematical objects that asserts the existence of a countable set with certain properties. Without such an axiom, such a set might not provably exist.
Important examples
Important countability axioms for topological spaces include:[1]
Relationships with each other
These axioms are related to each other in the following ways:
- Every first-countable space is sequential.
- Every second-countable space is first countable, separable, and Lindelöf.
- Every σ-compact space is Lindelöf.
- Every metric space is first countable.
- For metric spaces, second-countability, separability, and the Lindelöf property are all equivalent.
Other examples of mathematical objects obeying axioms of countability include sigma-finite measure spaces, and lattices of countable type.
References
